MATH 272 - Linear Algebra

1. Perform elementary row operations on a matrix to put it in row-echelon or reduced row-echelon form.
2. Apply the row reduction of an augmented matrix to solve systems of linear equations, as well as the equivalent vector and matrix forms for those systems.
3. Express the solution set of a nonhomogeneous system of linear equations in terms of a particular solution and the solution set of the corresponding homogeneous system.
4. Find the standard matrix of a linear transformation.
5. Perform the matrix operations of addition, scalar multiplication, and matrix multiplication.
6. Determine whether a given matrix is invertible.
7. Compute the inverse of a given nonsingular matrix.
8. Determine the dimension of and find a basis for a subspace of n-dimensional real space, given a spanning set.
9. Compute the determinant of a square matrix using row or column expansion.
10. Compute the determinant of a square matrix using properties of the determinant.
11. Apply Cramer’s rule to solve a system of linear equations.
12. Find bases for the null space and column space of a matrix, and for the kernel and range of a linear transformation.
13. Find coordinates of a vector with respect to a given basis.
14. Change the basis for a linear transformation.
15. Compute eigenvalues and eigenvectors for a matrix or linear transformation.
16. Diagonalize a transformation with distinct eigenvalues.
17. Use definitions and theorems to justify statements regarding any of the above concepts.